Galois Representations, -series and a Conjecture of Artin

Endow
with the topology which has as a basis of open neighborhoods
of the origin the subgroups
, where varies
over finite Galois extensions of
. (Note: This is **not** the
topology got by taking as a basis of open neighborhoods the collection
of finite-index normal subgroups of
.)
Fix a positive integer and let
be the group of
invertible matrices over
with the discrete topology.

Fix a Galois representation and let be the fixed field of , so factors through . For each prime that is not ramified in , there is an element that is well-defined up to conjugation by elements of . This means that is well-defined up to conjugation. Thus the characteristic polynomial of is a well-defined invariant of and . Let

We view as a function of a single complex variable . One can prove that is holomorphic on some right half plane, and extends to a meromorphic function on all .

The conjecture is known when . Assume for the rest of this
paragraph that is odd, i.e., if
is complex
conjugation, then
. When and the image of
in
is a solvable group, the conjecture is known,
and is a deep theorem of Langlands and others (see
[Lan80]), which played a crucial roll in Wiles's
proof of Fermat's Last Theorem. When and the image of in
is not solvable, the only possibility is that the
projective image is isomorphic to the alternating group .
Because is the symmetry group of the icosahedron, these
representations are called *icosahedral*. In this case, Joe
Buhler's Harvard Ph.D. thesis [Buh78] gave the first
example in which was shown to satisfy
Conjecture 9.5.3. There is a book [Fre94],
which proves Artin's conjecture for 7 icosahedral representation (none
of which are twists of each other). Kevin Buzzard and the author
proved the conjecture for 8 more examples [BS02].
Subsequently, Richard Taylor, Kevin Buzzard, Nick Shepherd-Barron, and
Mark Dickinson proved the conjecture for an infinite class of
icosahedral Galois representations (disjoint from the examples)
[BDSBT01]. The general problem for is in fact now completely
solved, due to recent work of Khare and Wintenberger
[KW08] that proves Serre's conjecture.

William Stein 2012-09-24